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Simplify the following: (x2+x-6)/(x2+7x+12) x (x-2)/(x(x+4))

Mathematics

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Answer

(x−2)2x(x+4)2\frac{(x-2)^2}{x(x+4)^2}

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Here is the solution to the question:

2.2.3. Simplify the following:

x2+x−6x2+7x+12×x−2x(x+4)\frac{x^2+x-6}{x^2+7x+12} \times \frac{x-2}{x(x+4)}

  1. Factorize the quadratic expressions in the first fraction.

    The numerator x2+x−6x^2+x-6 factors into (x+3)(x−2)(x+3)(x-2).
    The denominator x2+7x+12x^2+7x+12 factors into (x+3)(x+4)(x+3)(x+4).
    Substitute these factored forms back into the expression:

    (x+3)(x−2)(x+3)(x+4)×x−2x(x+4)\frac{(x+3)(x-2)}{(x+3)(x+4)} \times \frac{x-2}{x(x+4)}

  2. Cancel out the common factor (x+3)(x+3) from the first fraction, assuming x≠−3x \neq -3.

    x−2x+4×x−2x(x+4)\frac{x-2}{x+4} \times \frac{x-2}{x(x+4)}

  3. Multiply the numerators and the denominators.

    (x−2)(x−2)(x+4)x(x+4)\frac{(x-2)(x-2)}{(x+4)x(x+4)} (x−2)2x(x+4)2\frac{(x-2)^2}{x(x+4)^2}
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